Section 4.1

Solid Mechanics Part II Kelly 55

4.1 Cylindrical and Polar Coordinates 4.1.1 Geometrical Axisymmetry A large number of practical engineering problems involve geometrical features which have a natural axis of symmetry, such as the solid cylinder, shown in Fig. 4.1.1. The axis of symmetry is an axis of revolution; the feature which possesses axisymmetry (axial symmetry) can be generated by revolving a surface (or line) about this axis.

Figure 4.1.1: a cylinder Some other axisymmetric geometries are illustrated Fig. 4.1.2; a frustum, a disk on a shaft and a sphere.

Figure 4.1.2: axisymmetric geometries Some features are not only axisymmetric – they can be represented by a plane, which is similar to other planes right through the axis of symmetry. The hollow cylinder shown in Fig. 4.1.3 is an example of this plane axisymmetry.

axis of symmetry

create cylinder by revolving a surface

about the axis of symmetry

Section 4.1

Solid Mechanics Part II Kelly 56

Figure 4.1.3: a plane axisymmetric geometries Axially Non-Symmetric Geometries Axially non-symmetric geometries are ones which have a natural axis associated with them, but which are not completely symmetric. Some examples of this type of feature, the curved beam and the half-space, are shown in Fig. 4.1.4; the half-space extends to “infinity” in the axial direction and in the radial direction “below” the surface – it can be thought of as a solid half-cylinder of infinite radius. One can also have plane axially non-symmetric features; in fact, both of these are examples of such features; a slice through the objects perpendicular to the axis of symmetry will be representative of the whole object.

Figure 4.1.4: a plane axisymmetric geometries 4.1.2 Cylindrical and Polar Coordinates The above features are best described using cylindrical coordinates, and the plane versions can be described using polar coordinates. These coordinates systems are described next. Stresses and Strains in Cylindrical Coordinates Using cylindrical coordinates, any point on a feature will have specific ),,( zr θ coordinates, Fig. 4.1.5:

axisymmetric plane representative of

feature

Section 4.1

Solid Mechanics Part II Kelly 57

r – the radial direction (“out” from the axis) θ – the circumferential or tangential direction (“around” the axis –

counterclockwise when viewed from the positive z side of the 0=z plane) z – the axial direction (“along” the axis)

Figure 4.1.5: cylindrical coordinates The displacement of a material point can be described by the three components in the radial, tangential and axial directions. These are often denoted by

θuvuu r ≡≡ , and zuw ≡ respectively; they are shown in Fig. 4.1.6. Note that the displacement v is positive in the positive θ direction, i.e. the direction of increasing θ .

Figure 4.1.6: displacements in cylindrical coordinates The stresses acting on a small element of material in the cylindrical coordinate system are as shown in Fig. 4.1.7 (the normal stresses on the left, the shear stresses on the right).

plane0=z

r

z

θ

r

z

u

w

v

Section 4.1

Solid Mechanics Part II Kelly 58

Figure 4.1.7: stresses in cylindrical coordinates The normal strains θθεε ,rr and zzε are a measure of the elongation/shortening of material, per unit length, in the radial, tangential and axial directions respectively; the shear strains zr θθ εε , and zrε represent (half) the change in the right angles between line elements along the coordinate directions. The physical meaning of these strains is illustrated in Fig. 4.1.8.

Figure 4.1.8: strains in cylindrical coordinates Plane Problems and Polar Coordinates The stresses in any particular plane of an axisymmetric body can be described using the two-dimensional polar coordinates ( )θ,r shown in Fig. 4.1.9.

strain at point o

rrε = unit elongation of oA

θθε = unit elongation of oB

zzε = unit elongation of oC

θε r = ½ change in angle AoB∠

zθε = ½ change in angle BoC∠

zrε = ½ change in angle AoC∠

r

θ

z

A

B

C

o

rrσrrσ

zzσ

zzσ

θθσ

θθσ

θσ r

zθσzrσ

Section 4.1

Solid Mechanics Part II Kelly 59

Figure 4.1.9: polar coordinates There are three stress components acting in the plane 0=z : the radial stress rrσ , the circumferential (tangential) stress θθσ and the shear stress θσ r , as shown in Fig. 4.1.10. Note the direction of the (positive) shear stress – it is conventional to take the z axis out of the page and so the θ direction is counterclockwise. The three stress components which do not act in this plane, but which act on this plane ( zzz θσσ , and zrσ ), may or may not be zero, depending on the particular problem (see later).

Figure 4.1.10: stresses in polar coordinates

rrσ

rrσθθσ

θσ r

θθσθσ r

θr